We consider convolution operators on $\mathbb{R}^n$ of the form $$T_Pf(x) =\int_{\mathbb{R}^m} f\big(x-P(y)\big)K(y) dy$$, where $P$ is a polynomial defined on $\mathbb{R}^m$ with values in $\mathbb{R}^n$ and $K$ is a smooth Calderón-Zygmund kernel on $\mathbb{R}^m$. A maximal operator $M_P$ can be constructed in a similar fashion. We discuss weak-type 1-1 estimates for $T_P$ and $M_P$ and the uniformity of such estimates with respect to $P$. We also obtain $L^p$-estimates for "supermaximal" operators, defined by taking suprema over $P$ ranging in certain classes of polynomials of bounded degree.

We extend an uncertainty principle due to Beurling into a characterization of Hermite functions. More precisely, all functions $f$ on $\mathbb{R}^d$ which may be written as $P(x)\exp (-\langle Ax, x\rangle)$, with $A$ a real symmetric definite positive matrix, are characterized by integrability conditions on the product $f(x) \widehat{f}(y)$. We then obtain similar results for the windowed Fourier transform (also known, up to elementary changes of functions, as the radar ambiguity function or the Wigner transform). We complete the paper with a sharp version of Heisenberg's inequality for this transform.

We prove the existence of complex geometrical optics solutions for Lipschitz conductivities. Moreover we show that, in dimensions $n\ge 3$ that one can uniquely recover a $W^{3/2, \infty}$ conductivity from its associated Dirichlet-to-Neumann map or voltage to current map.

We provide a detailed treatment of the Camassa-Holm (CH) hierarchy with special emphasis on its algebro-geometric solutions. In analogy to other completely integrable hierarchies of soliton equations such as the KdV or AKNS hierarchies, the CH hierarchy is recursively constructed by means of a basic polynomial formalism invoking a spectral parameter. Moreover, we study Dubrovin-type equations for auxiliary divisors and associated trace formulas, consider the corresponding algebro-geometric initial value problem, and derive the theta function representations of algebro-geometric solutions of the CH hierarchy.

The notion of the extremal length and the module of families of curves has been studied extensively and has given rise to a lot of applications to complex analysis and the potential theory. In particular, the coincidence of the $p$-module and the $p$-capacity plays an important role. We consider this problem on the Carnot group. The Carnot group $\mathbb{G}$ is a simply connected nilpotent Lie group equipped with an appropriate family of dilations. Let $\Omega$ be a bounded domain on $\mathbb{G}$ and $K_0$, $K_1$ be disjoint non-empty compact sets in the closure of $\Omega$. We consider two quantities, associated with this geometrical structure $(K_0,K_1;\Omega)$. Let $M_p(\Gamma(K_0,K_1;\Omega))$ stand for the $p$-module of a family of curves which connect $K_0$ and $K_1$ in $\Omega$. Denoting by $\cap_p(K_0,K_1;\Omega)$ the $p$-capacity of $K_0$ and $K_1$ relatively to $\Omega$, we show that $$M_p(\Gamma(K_0,K_1;\Omega))=\cap_p(K_0,K_1;\Omega)$$.

We prove that the 3D cubic defocusing semi-linear wave equation is globally well-posed for data in the Sobolev space $\dot{H}^{s}$ where $s>3/4$. This result was obtained in [Kenig-Ponce-Vega, 2000] following Bourgain's method ([Bourgain, 1998]). We present here a different and somewhat simpler argument, inspired by previous work on the Navier-Stokes equations ([Calderon, 1990], [Gallagher-Planchon, 2002])

In this paper we study the Cauchy problem for the nonlinear Dirac equation in the Sobolev space $H^s$. We prove the existence and uniqueness of global solutions for small data in $H^s$ with $s>1$. The method of proof is based on the Strichartz estimate of $L^2_t$ type for Dirac and Klein-Gordon equations. We also prove that the solutions of the nonlinear Dirac equation after modulation of phase converge to the corresponding solutions of the nonlinear Schröodinger equation as the speed of light tends to infinity.

We prove, for arbitrary dimension of the base $n\ge 4$, that stationary Yang-Mills Fields satisfying some approximability property are regular apart from a closed subset of the base having zero $(n-4)$-Hausdorff measure.

We prove that the elliptic maximal function maps the Sobolev space $W_{4,\eta}(\mathbb{R}^2)$ into $L^4(\mathbb{R}^2)$ for all $\eta>1/6$. The main ingredients of the proof are an analysis of the intersection properties of elliptic annuli and a combinatorial method of Kolasa and Wolff.

We establish new results on the space BV of functions with bounded variation. While it is well known that this space admits no unconditional basis, we show that it is "almost" characterized by wavelet expansions in the following sense: if a function $f$ is in BV, its coefficient sequence in a BV normalized wavelet basis satisfies a class of weak-$\ell^1$ type estimates. These weak estimates can be employed to prove many interesting results. We use them to identify the interpolation spaces between BV and Sobolev or Besov spaces, and to derive new Gagliardo-Nirenberg-type inequalities.